Students' difficulties with vector calculus in electrodynamics

Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell's equations in electrodynamics.Comment: 14 pages, 11 figure

Student thinking about the divergence and curl in mathematics and physics contexts

Undergraduate physics students are known to have difficulties with understanding mathematical tools, and with applying their knowledge of mathematics to physical contexts. Using survey statements based on student interviews and written responses to open-ended questions, we investigated the prevalence of correct and incorrect conceptions regarding the divergence and curl of vector fields, among both mathematics and physics students. We compare and contrast pre-instruction responses from intermediate-level E&M students at KU Leuven and the University of St Andrews, with post-instruction responses from St Andrews students enrolled in a vector calculus course. The differences between these student populations were primarily in areas having to do with physics-related concepts and graphical representations of vector fields. Our comparison of pre- and post-instruction responses from E&M students shows that their understanding of the divergence and curl improved significantly in most areas, though not as much as would be desired.Comment: Physics Education Research Conference 2015 (submitted

Stochastic Lag Time in Nucleated Linear Self-Assembly

Protein aggregation is of great importance in biology, e.g., in amyloid fibrillation. The aggregation processes that occur at the cellular scale must be highly stochastic in nature because of the statistical number fluctuations that arise on account of the small system size at the cellular scale. We study the nucleated reversible self-assembly of monomeric building blocks into polymer-like aggregates using the method of kinetic Monte Carlo. Kinetic Monte Carlo, being inherently stochastic, allows us to study the impact of fluctuations on the polymerisation reactions. One of the most important characteristic features in this kind of problem is the existence of a lag phase before self-assembly takes off, which is what we focus attention on. We study the associated lag time as a function of the system size and kinetic pathway. We find that the leading order stochastic contribution to the lag time before polymerisation commences is inversely proportional to the system volume for large-enough system size for all nine reaction pathways tested. Finite-size corrections to this do depend on the kinetic pathway

Qualitative investigation into students’ use of divergence and curl in electromagnetism

Many students struggle with the use of mathematics in physics courses. Although typically well trained in rote mathematical calculation, they often lack the ability to apply their acquired skills to physical contexts. Such student difficulties are particularly apparent in undergraduate electrodynamics, which relies heavily on the use of vector calculus. To gain insight into student reasoning when solving problems involving divergence and curl, we conducted eight semistructured individual student interviews. During these interviews, students discussed the divergence and curl of electromagnetic fields using graphical representations, mathematical calculations, and the differential form of Maxwell’s equations. We observed that while many students attempt to clarify the problem by making a sketch of the electromagnetic field, they struggle to interpret graphical representations of vector fields in terms of divergence and curl. In addition, some students confuse the characteristics of field line diagrams and field vector plots. By interpreting our results within the conceptual blending framework, we show how a lack of conceptual understanding of the vector operators and difficulties with graphical representations can account for an improper understanding of Maxwell’s equations in differential form. Consequently, specific learning materials based on a multiple representation approach are required to clarify Maxwell’s equations.Publisher PDFPeer reviewe

Anomalous kinetics of attractive A+B→0A+B \to 0 reactions

We investigate the kinetics of $A+B \to 0$ reaction with the local attractive interaction between opposite species in one spatial dimension. The attractive interaction leads to isotropic diffusions inside segregated single species domains, and accelerates the reactions of opposite species at the domain boundaries. At equal initial densities of $A$ and $B$, we analytically and numerically show that the density of particles ($\rho$), the size of domains ($\ell$), the distance between the closest neighbor of same species ($\ell_{AA}$), and the distance between adjacent opposite species ($\ell_{AB}$) scale in time as $\rho \sim t^{-1/3}$, $\ell_{AA} \sim t^{1/3}$, and $\ell \sim \ell_{AB} \sim t^{2/3}$ respectively. These dynamical exponents form a new universality class distinguished from the class of uniformly driven systems of hard-core particles.Comment: 4 pages, 4 figure

4p-inner-shell and double-excitation spectrum of Sr II

We report photoabsorption measurements from the 4p inner shell of singly ionized strontium in the spectral region between 26.0 and 37.4 eV. More than 60 levels of Sr II are reported which are interpreted as singly excited inner-shell transitions 4p65s 2S1∕2→4p45s(1P,3P)ns, md and doubly excited transitions 4p65s 2S1∕2→4p54d(1P,3P,3D)nl. Multiconfiguration Hartree-Fock calculations are performed in jj coupling and the levels are arranged into Rydberg series converging on seven limits, allowing the identification of two levels in Sr III

Dificultades de estudiantes universitarios de tres países en el aprendizaje del concepto de fuerza electromotriz en electricidad

El objetivo principal de este estudio es identificar las dificultades de estudiantes universitarios en el aprendizaje de los conceptos de fuerza electromotriz y diferencia de potencial eléctrico en el contexto de corrientes transitorias y de circuitos de corriente directa con resistencias. Para investigar las dificultades de los estudiantes desarrollamos un cuestionario basado en un análisis del marco teórico y epistemológico de la física. Este cuestionario se aplicó a estudiantes de primer año de ingeniería y de física de universidades en el País Vasco, Colombia y Bélgica